RAS question
How many rectangles (including squares) are there in a figure formed by 4 horizontal and 3 vertical lines?
Correct answer: (B) 18.
A figure formed by 4 horizontal and 3 vertical lines contains 18 rectangles, including squares.
Explanation
Each rectangle in the grid is fixed by choosing two horizontal boundary lines and two vertical boundary lines. NCERT's combinations rule applies because this is a selection problem: the order in which the boundary lines are chosen does not create a different rectangle. From 4 horizontal lines, the number of possible pairs is C(4,2) = 6. From 3 vertical lines, the number of possible pairs is C(3,2) = 3. Pairing each horizontal choice with each vertical choice gives 6 x 3 = 18 rectangles. Squares are already included because every square is also a rectangle; the counting method selects boundary lines, not shapes by appearance.
Why the other options are wrong
- (A) 12 undercounts the figure because it does not use all 6 possible pairs of horizontal lines with all 3 possible pairs of vertical lines.
- (C) 24 is too high because the correct combination count is C(4,2) x C(3,2), not a larger product.
- (D) 15 misses three valid rectangles formed by selecting different pairs of horizontal and vertical boundary lines.
Concept
This tests combinations in reasoning: selecting boundary lines where order is immaterial. RAS asks such grid-counting problems because they quickly test whether a candidate can convert a visual arrangement into a counting formula.